Hi! Firstly, let me apologize for the 1 month and half hiatus: while I plan to close up my series on the talks and courses of the Dynamical Systems conference (held at ICTP, Italy), I wasn’t able to do so during April because I was learning some interesting facts about square-tiled surfaces and Teichmuller dynamics from my new friends Pascal Hubert and Anton Zorich. Of course, I plan to share these facts in some future posts, although I can’t promise to do so in the next few months. On the other hand, I can give a clue to the curious reader by providing a copy of the slides of my recent talk at Orsay entitled: Sur le cocycle de Kontsevich-Zorich au-dessus de deux origamis spéciaux.
Warning. While the title is written in French (and the actual talk was my third presentation speaking in French), the slides are written in English (because I don’t feel comfortable to write in French yet).
Before giving the link to the slides, let me make a brief description of the talk.
The basic objective was the discussion of the Teichmuller geodesic flow and the Kontsevich-Zorich cocycle. Nowadays, after the works of Forni (2002), Avila and Viana (2006), we know that the Kontsevich-Zorich conjecture is true: the Lyapunov spectrum of the Kontsevich-Zorich (KZ) cocycle with respect to the absolutely continuous -invariant ergodic probability is non-uniformly hyperbolic (i.e., 0 doesn’t belong to the spectrum) and simple (i.e., all exponents have multiplicity one). In other words, the ergodic behavior of generic points with respect to the Lebesgue measure is more or less well-understood.
However, one can follow Veech and pose the question: what about the ergodic behavior of non-generic points? In this direction, Forni (2005) and Forni, Matheus (2008) showed that there are examples of orbits with totally degenerate Kontsevich-Zorich spectrum (see this previous post). An interesting feature of the KZ cocycle over totally degenerate orbits is the fact that it is isometric (this follows from Forni’s 2002 work). In the talk, I discussed an improvement of this fact (obtained jointly with Jean-Christophe Yoccoz): by computing the action of the group of affine diffeomorphisms on the relative homology of these examples, we are able to show that the Kontsevich-Zorich cocycle is represented by a subgroup of the Weyl group of the root system . A nice consequence is the fact that the KZ cocycle acts by a finite group of rigid matrices, so that this gives a new proof of Forni and Matheus results by more explicit methods.
Finally, the slides can be found here.